| 1. |
- Ahlberg, Daniel, 1982-, et al.
(author)
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Annihilating Branching Brownian Motion
- 2024
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In: International mathematics research notices. - 1073-7928 .- 1687-0247. ; 2024:13, s. 10425-10448
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Journal article (peer-reviewed)abstract
- We study an interacting system of competing particles on the real line. Two populations of positive and negative particles evolve according to branching Brownian motion. When opposing particles meet, their charges neutralize and the particles annihilate, as in an inert chemical reaction. We show that, with positive probability, the two populations coexist and that, on this event, the interface is asymptotically linear with a random slope. A variety of generalizations and open problems are discussed.
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| 2. |
- Ahlberg, Daniel, 1982
(author)
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Asymptotics and dynamics in first-passage and continuum percolation
- 2011
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Doctoral thesis (other academic/artistic)abstract
- This thesis combines the study of asymptotic properties of percolation processes with various dynamical concepts. First-passage percolation is a model for the spatial propagation of a fluid on a discrete structure; the Shape Theorem describes its almost sure convergence towards an asymptotic shape, when considered on the square (or cubic) lattice. Asking how percolation structures are affected by simple dynamics or small perturbations presents a dynamical aspect. Such questions were previously studied for discrete processes; here, sensitivity to noise is studied in continuum percolation. Paper I studies first-passage percolation on certain 1-dimensional graphs. It is found that when identifying a suitable renewal sequence, its asymptotic behaviour is much better understood compared to higher dimensional cases. Several analogues of classical 1-dimensional limit theorems are derived. Paper II is dedicated to the Shape Theorem itself. It is shown that the convergence, apart from holding almost surely and in L^1, also holds completely. In addition, inspired by dynamical percolation and dynamical versions of classical limit theorems, the almost sure convergence is proved to be dynamically stable. Finally, a third generalization of the Shape Theorem shows that the above conclusions also hold for first-passage percolation on certain cone-like subgraphs of the lattice. Paper III proves that percolation crossings in the Poisson Boolean model, also known as the Gilbert disc model, are noise sensitive. The approach taken generalizes a method introduced by Benjamini, Kalai and Schramm. A key ingredient in the argument is an extremal result on arbitrary hypergraphs, which is used to show that almost no information about the critical process is obtained when conditioning on a denser Poisson process.
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| 3. |
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| 4. |
- Ahlberg, Daniel, 1982
(author)
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Asymptotics of First-Passage Percolation on One-Dimensional Graphs
- 2015
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In: Advances in Applied Probability. - : Cambridge University Press (CUP). - 0001-8678 .- 1475-6064. ; 47:1, s. 182-209
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Journal article (peer-reviewed)abstract
- In this paper we consider first-passage percolation on certain one-dimensional periodic graphs, such as the Z x {0, 1, ..., K - 1}(d-1) nearest neighbour graph for d, K >= 1. We expose a regenerative structure within the first-passage process, and use this structure to show that both length and weight of minimal-weight paths present a typical one-dimensional asymptotic behaviour. Apart from a strong law of large numbers, we derive a central limit theorem, a law of the iterated logarithm, and a Donsker theorem for these quantities. In addition, we prove that the mean and variance of the length and weight of minimizing paths are monotone in the distance between their end-points, and further show how the regenerative idea can be used to couple two first-passage processes to eventually coincide. Using this coupling we derive a 0-1 law.
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| 5. |
- Ahlberg, Daniel, 1982, et al.
(author)
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Bernoulli and self-destructive percolation on non-amenable graphs
- 2014
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In: Electronic Communications in Probability. - 1083-589X. ; 19, s. article 40-
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Journal article (peer-reviewed)abstract
- In this note we study some properties of infinite percolation clusters on non-amenable graphs. In particular, we study the percolative properties of the complement of infinite percolation clusters. An approach based on mass-transport is adapted to show that for a large class of non-amenable graphs, the graph obtained by removing each site contained in an infinite percolation cluster has critical percolation threshold which can be arbitrarily close to the critical threshold for the original graph, almost surely, as p approaches p_c. Closely related is the self-destructive percolation process, introduced by J. van den Berg and R. Brouwer, for which we prove that an infinite cluster emerges for any small reinforcement.
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| 6. |
- Ahlberg, Daniel, 1982
(author)
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Convergence Towards an Asymptotic Shape in First-Passage Percolation on Cone-Like Subgraphs of the Integer Lattice
- 2015
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In: Journal of Theoretical Probability. - : Springer Science and Business Media LLC. - 0894-9840 .- 1572-9230. ; 28:1, s. 198-222
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Journal article (peer-reviewed)abstract
- In first-passage percolation on the integer lattice, the shape theorem provides precise conditions for convergence of the set of sites reachable within a given time from the origin, once rescaled, to a compact and convex limiting shape. Here, we address convergence towards an asymptotic shape for cone-like subgraphs of the lattice, where . In particular, we identify the asymptotic shapes associated with these graphs as restrictions of the asymptotic shape of the lattice. Apart from providing necessary and sufficient conditions for - and almost sure convergence towards this shape, we investigate also stronger notions such as complete convergence and stability with respect to a dynamically evolving environment.
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| 7. |
- Ahlberg, Daniel, 1982-, et al.
(author)
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From stability to chaos in last-passage percolation
- 2024
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In: Bulletin of the London Mathematical Society. - 0024-6093 .- 1469-2120. ; 56:1, s. 411-422
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Journal article (peer-reviewed)abstract
- We study the transition from stability to chaos in a dynamic last passage percolation model on with random weights at the vertices. Given an initial weight configuration at time 0, we perturb the model over time in such a way that the weight configuration at time t is obtained by resampling each weight independently with probability t. On the cube [0, n]d, we study geodesics, that is, weight-maximizing up-right paths from (0,0,⋯,0) to (n,n,⋯,n), and their passage time T. Under mild conditions on the weight distribution, we prove a phase transition between stability and chaos at t≍ Var(T). Indeed, as n grows large, for small values of t, the passage times at time 0 and time t are highly correlated, while for large values of t, the geodesics become almost disjoint.
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| 8. |
- Ahlberg, Daniel, 1982, et al.
(author)
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Noise sensitivity in continuum percolation
- 2014
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In: Israel Journal of Mathematics. - : Springer Science and Business Media LLC. - 0021-2172 .- 1565-8511. ; 201:2, s. 847-899
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Journal article (peer-reviewed)abstract
- We prove that the Poisson Boolean model, also known as the Gilbert disc model, is noise sensitive at criticality. This is the first such result for a Continuum Percolation model, and the first which involves a percolation model with critical probability pc not equal 1/2. Our proof uses a version of the Benjamini-Kalai-Schramm Theorem for biased product measures. A quantitative version of this result was recently proved by Keller and Kindler. We give a simple deduction of the non-quantitative result from the unbiased version. We also develop a quite general method of approximating Continuum Percolation models by discrete models with pc bounded away from zero; this method is based on an extremal result on non-uniform hypergraphs.
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| 9. |
- Ahlberg, Daniel, 1982, et al.
(author)
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Scaling limits for the threshold window: When does a monotone Boolean function flip its outcome?
- 2017
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In: Annales de linstitut Henri Poincare (B) Probability and Statistics. - 0246-0203 .- 1778-7017. ; 53:4, s. 2135-2161
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Journal article (peer-reviewed)abstract
- Consider a monotone Boolean function f:{0,1}^n \to {0,1} and the canonical monotone coupling {eta_p:p in [0,1]} of an element in {0,1}^n chosen according to product measure with intensity p in [0,1]. The random point p in [0,1] where f(eta_p) flips from 0 to 1 is often concentrated near a particular point, thus exhibiting a threshold phenomenon. For a sequence of such Boolean functions, we peer closely into this threshold window and consider, for large n, the limiting distribution (properly normalized to be nondegenerate) of this random point where the Boolean function switches from being 0 to 1. We determine this distribution for a number of the Boolean functions which are typically studied and pay particular attention to the functions corresponding to iterated majorityand percolation crossings. It turns out that these limiting distributions have quite varying behavior. In fact, we show that any nondegenerate probability measure on R arises in this way for some sequence of Boolean functions.
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| 10. |
- Elgqvist, Jörgen, 1963, et al.
(author)
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Intraperitoneal Alpha-Radioimmunotherapy of Advanced Ovarian Cancer in Nude Mice using Different High Specific Activities
- 2010
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In: World Journal of Oncology. - : Elmer Press, Inc.. - 1920-4531. ; 1:3, s. 101-110
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Journal article (peer-reviewed)abstract
- Background: The aim of this study was to investigate the therapeutic efficacy of advanced ovarian cancer in mice, using α-radioimmunotherapy with different high specific activities. The study was performed using the monoclonal antibody (mAb) MX35 F(ab´)2 labeled with the α-particle emitter 211At.Methods: Animals were intraperitoneally inoculated with ≥1 × 107 cells of the ovarian cancer cell line NIH:OVCAR-3. Four weeks later 9 groups of animals were given 25, 50, or 400 kBq 211At-MX35 F(ab´)2 with specific activities equal to 1/80, 1/500, or 1/1200 (211At atom/number of mAbs) for every activity level respectively (n = 10 in each group). As controls, animals were given PBS or unlabeled MX35 F(ab´)2 in PBS (n = 10 in each group). Eight weeks after treatment the animals were sacrificed and the presence of macroscopic tumors was determined by meticulous ocular examination of the abdominal cavity. Cumulated activity and absorbed dose calculations on tumor cells and tumors were performed using in house developed program. Specimens for scanning electron-microscopy analysis were collected from the peritoneum at the time of dissection.Results: Summing over the different activity levels (25, 50, and 400 kBq 211At-MX35 F(ab´)2) the number of animals with macroscopic tumors was 13, 17, and 22 (n = 30 for each group) for the specific activities equal to 1/80, 1/500, or 1/1200, respectively. Logistic-regression analysis showed a significant trend that higher specific activity means less probability for macroscopic tumors (P = 0.02).Conclusions: Increasing the specific activity indicates a way to enhance the therapeutic outcome of advanced ovarian cancer, regarding macroscopic tumors. Further studies of the role of the specific activity are therefore justified.
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